Has anyone solved Claude Artifact authentication inside an embedded website?

I'm experimenting with embedding a published Claude Artifact into an external website. The Artifact displays correctly, but its AI functionality doesn't seem to recognise an existing Claude login session. When I authenticate through the prompted login flow, the Artifact works in a separate browser tab, but the embedded version continues requesting authentication. I've tested this across multiple browsers with similar results.

Has anyone encountered and resolved this?

I'm particularly interested in whether this is a browser authentication limitation, an embedding configuration issue, or a known restriction with AI-powered Claude Artifacts. Would appreciate hearing from anyone who has successfully implemented this or found an officially supported workaround.